情報ネットワーク科学研究会 非線形振動子の同期と その実験について 埼玉大学 大学院理工学研究科 数理電子情報部門 埼玉大学 脳科学融合研究センター 池口 徹.

Transcription

1 情報ネットワーク科学研究会 非線形振動子の同期と その実験について 埼玉大学 大学院理工学研究科 数理電子情報部門 埼玉大学 脳科学融合研究センター 池口 徹

2 振り子時計 クリスチャン ホイヘンス ( ) 2

3 同期の発見 1665年2月 A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universtal Concept in Nonlinear Science, Cambridge University Press,

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9 Synchronization of metronomes James Pantaleone Department of Physics, University of Alaska, Anchorage, Alaska Received 1 April 2002; accepted 24 June 2002 Synchronization is a common phenomenon in physical and biological systems. We examine the synchronization of two and more metronomes placed on a freely moving base. The small motion of the base couples the pendulums causing synchronization. The synchronization is generally in-phase, with antiphase synchronization occurring only under special conditions. The metronome system provides a mechanical realization of the popular Kuramoto model for synchronization of biological oscillators, and is excellent for classroom demonstrations and an undergraduate physics lab American Association of Physics Teachers. DOI: / I. INTRODUCTION AND SUMMARY Synchronization is the process where two or more systems interact with each other and come to move together. It is commonly observed to occur between oscillators. Synchronization differs from the well-known phenomena of resonance, where an oscillator responds to an external periodic signal. Collections of oscillators are observed to synchronize in a diverse variety of systems, despite the inevitable differences between the oscillators. Synchronization is a fundamental theme in nonlinear phenomena and is currently a popular topic of research. 1 Biology abounds with examples of synchronization. 2,3 Populations of certain cicada species emerge simultaneously with periods of 13 or 17 years. 4 Huge swarms of fireflies in South-East Asia gather in the same tree to flash in synchrony see, for example, Ref. 5. Networks of pacemaker cells in the heart beat together. 6 An example from psychology is the synchronization of clapping in audiences. 7 There are many physical examples also. The voltage oscillations of superconducting Josephson junctions are observed to synchronize. 8,9 Here we examine a variant of Huygens original system, two pendulum metronomes on a light, easily movable platform. For small intrinsic frequency differences, the oscillators generally synchronize with a small phase difference, that is, in-phase. This system makes an excellent classroom demonstration: it can be assembled quickly, synchronization occurs in a few tens of seconds, the mechanical motion is visually appealing, and the metronomes ticks provide an added indication of the pendulum bob s motion. The system is also useful for an experimental study of synchronization. The audible ticks and/or the base motion provide an easy way to quantify the relative motion of the pendulum bobs. The ticks can be recorded and used to study the approach to synchronization and the small phase difference between the synchronized metronomes. The results are well described by a simple model. The metronomes are described as van der Pol oscillators 17 and the coupling between the metronomes comes from the undamped motion of the base. Using this model, the absence of the antiphase synchronization that Huygens and others observed is readily explained. The large oscillation amplitudes 11

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12 Vol November 2005 BRIEF COMMUNICATIONS Crowd synchrony on the Millennium Bridge Footbridges start to sway when packed with pedestrians falling into step with their vibrations. Soon after the crowd streamed on to London s Millennium Bridge on the day it opened, the bridge started to sway from side to side: many pedestrians fell spontaneously into step with the bridge s vibrations, inadvertently amplifying them. Here we model this unexpected and now notorious phenomenon which was not due to the bridge s innovative design as was first thought by adapting ideas originally developed to describe the collective synchronization of biological oscillators such as neurons and fireflies. Our approach should help engineers to estimate the damping needed to stabilize other exceptionally crowded footbridges against synchronous lateral excitation by pedestrians. Existing theories 1 6 of what happened on the bridge s opening day focus on the wobbling of the bridge but have not addressed the X /Ω 0 Ψ A X B K M G cos Θ i Θ i G sin Θ i X G sin Θ i Figure 1 Effect of pedestrian crowding on London s Millennium Bridge. The resonant lateral mode of vibration of the bridge (left) can be represented by a mass-spring-damper system (top, right). The angular phases (bottom) for the bridge displacement X (left) and the individual pedestrian forces, Gsin i (right), are indicated (see text for definitions of variables). X G instability until the crowd reaches a critical size, N c, after which wobbling and synchrony erupt simultaneously (Fig. 2b, c). We can calculate N c analytically, using methods 8 10 created to study large systems of biological oscillators (see supplementary information). To take the simplest case, suppose /2 and P( ) is symmetrical about 0 (also a worst case for the bridge, in the sense that pedestrians then drive it most efficiently). We find 4 K N c (3) GC P( 0 ) where B/ 4MK is the damping ratio. All the parameters have known values, except for C. Comparing our simulations with data obtained from crowd tests on the Millennium Bridge 2, we estimate C 16 m 1 s 1. Then, with no fur- 17

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14 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 22, (1993) SYNCHRONIZATION OF HUMAN WALKING OBSERVED DURING LATERAL VIBRATION OF A CONGESTED PEDESTRIAN BRIDGE YOZO FUJINO* Department of Civil Engineering, The University of Tokyo, Tokyo 113. Japan BENITO. M. PACHECO Ammann and Whitney Consulting Engineers, New York. NY , U.S.A SHUN-ICHI NAKAM URA * Bridge Engineering and Construction Division, Nippon Steel Corporation, Tokyo 100, Japan AND PENNUNG WARNITCHAI Division of Structural Engineering and Construction, Asian Institute of Technology, Bangkok, Thailand SUMMARY Observation of human-induced large-amplitude lateral vibration of an actual pedestrian bridge in an extremely congested condition is reported. Walking motions of pedestrians recorded by a video camera are analysed. It is found that walking among 20 per cent or more of the pedestrians on the bridge was synchronized to the girder lateral vibration. With this synchronization, the total lateral force from the pedestrians to the girder is evidently increased and it acts as a resonant force on the girder lateral vibration. INTRODUCTION 19

15 Figure 3. Congested condition Y. Fujino et al, Earthquake Engineering and Structural Dynamics, 2, ,

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17 24 Yoshida et al., 2006 Teramae & Tanaka, 2004 Arai & Nakao, 2008 Teramae & Tanaka, 2006 Nagai & Nakao, 2009 Kurebayashi, Fujiwara & Ikeguchi, 2012 Kawai, Fujiwara, Jinno, Horio & Ikeguchi, 2012, 2013

18 25 [Z. F. Mainen and T. J. Sejnowski, Science (1995)]

19 26 [A. B. Neiman et al., Phys. Rev. Lett.(2002)]

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21 V out V in 30

22 V out = V max (V + > V ) 0 (V + V ) V V + R 2 C 1 dv dt = V + E (V + > V ) R 2 C 1 dv dt = V (V + V ) V t V V out V + V max 0 t31

23 32 Teramae & Tanaka, 2004 Teramae & Tanaka, 2006 Kurebayashi, Fujiwara & Ikeguchi, 2012 Arai & Nakao, 2008 Nagai & Nakao, 2009 Kawai, Fujiwara, Jinno, Horio & Ikeguchi, 2012, 2013 Zhou & Kurths, 2002 Wang, Lai & Zheng, 2009 Uchida, McAllister & Roy, 2004 Kawai, Shimada, Fujiwara & Ikeguchi, in preparation.

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25 メトロノーム同期の分岐 47